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Difference between revisions of "2006 IMO Problems/Problem 6"

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Assign to each side <math>b</math> of a convex polygon <math>P</math> the maximum area of a triangle that has <math>b</math> as a side and is contained in <math>P</math>. Show that the sum of the areas assigned to the sides of <math>P</math> is at least twice the area of <math>P</math>.
 
Assign to each side <math>b</math> of a convex polygon <math>P</math> the maximum area of a triangle that has <math>b</math> as a side and is contained in <math>P</math>. Show that the sum of the areas assigned to the sides of <math>P</math> is at least twice the area of <math>P</math>.
  
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==Solution==
 
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{{solution}}
 
  
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<math>MERLIN</math> HAS INFINITE CHARISMA
 
  
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<math>MERLIN IS A SIGMA</math>
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==Solution==
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Merlin (also called Emrys) is the hero and protagonist of the series. He is a warlock, the only son of Hunith and Balinor, the ward and apprentice of Gaius, and the best friend and manservant of the late King Arthur. He is also the last Dragonlord in existence and a creature of the Old Religion.
 
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After leaving his childhood home for Camelot, Merlin became the manservant of Prince Arthur. From then on, Merlin began to protect and guide Arthur on his journey to the throne; according to the Great Dragon, it was his destiny to do so. Merlin was also destined to become the greatest and most powerful sorcerer to ever live, and to use his powers to help Arthur reunite the kingdom of Albion by uniting the Old Ways with the new.
 
  
 
==See Also==
 
==See Also==
  
 
{{IMO box|year=2006|num-b=5|after=Last Problem}}
 
{{IMO box|year=2006|num-b=5|after=Last Problem}}

Latest revision as of 03:20, 31 August 2024

Problem

Assign to each side $b$ of a convex polygon $P$ the maximum area of a triangle that has $b$ as a side and is contained in $P$. Show that the sum of the areas assigned to the sides of $P$ is at least twice the area of $P$.


Solution

See Also

2006 IMO (Problems) • Resources
Preceded by
Problem 5 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Last Problem
All IMO Problems and Solutions
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